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Constant mean curvature graphs in $\mathbb{H}^3$ defined in exterior domains

Patricia Klaser, Adilson Nunes, Jaime Ripoll

Abstract

Given $H\in [0,1)$ and given a $C^0$ exterior domain $Ω$ in a $H-$hypersphere of $\mathbb{H}^3,$ the existence of hyperbolic Killing graphs of CMC $H$ defined in $\overlineΩ$ with boundary $ \partial Ω$ included in the $H-$hypersphere is obtained.

Constant mean curvature graphs in $\mathbb{H}^3$ defined in exterior domains

Abstract

Given and given a exterior domain in a hypersphere of the existence of hyperbolic Killing graphs of CMC defined in with boundary included in the hypersphere is obtained.

Paper Structure

This paper contains 4 sections, 11 theorems, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Rotational surfaces
  3. Main results
  4. The case $H\in (0,1)$

Key Result

Theorem 1.1

Let $X$ be a hyperbolic Killing field of $\mathbb{H}^{n+1}$ orthogonal to a totally geodesic hypersurface $\mathbb{H}^n.$ Let $\Omega \subset \mathbb{H}^n$ be a $C^0$ exterior domain, that is, $\mathbb{H}^n\backslash \Omega$ is compact. Then, given $s\ge 0,$ there is $u\in C^\infty(\Omega)\cap C^0(\

Figures (3)

  • Figure 1: Domain $\Omega$ in $E_H$
  • Figure 2: Maximum principle
  • Figure 3: Relating ${\rm grad\, } u$ and ${\rm grad\, } \tilde{u}$

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof : Proof of Theorem \ref{['teo-H=0']}
  • Theorem 3.1
  • ...and 10 more